**Speaker:** Russell Lyons (Indiana University)

**Abstract:** Consider a Cayley graph of a group, \$\Gamma\$. Suppose that \$W\$ is a random assignment of nonnegative numbers to the edges and that the law of \$W\$ is \$\Gamma\$-invariant. Let \$X_t\$ be continuous-time random walk on \$\Gamma\$ in the random environment \$W\$: incident edges \$e\$ are crossed at rate \$W(e)\$. Write \$p^W(t) := {\bf E}\bigl[{\bf P}_o[X_t = o]\bigr]\$ for the expected return probability at time \$t\$ (averaged over \$W\$). Fontes and Mathieu asked whether given two such environments, \$W_1\$ and \$W_2\$, with \$W_1(e) \le W_2(e)\$ for all edges \$e\$, one has \$p^{W_1}(t) \ge p^{W_2}(t)\$ for all \$t \ge 0\$. When the pair \$(W_1, W_2)\$ has a \$\Gamma\$-invariant law, this was shown by Aldous and the speaker. It remains open in general. We attempt to attack this problem via similar questions for finite graphs.